Find the limits: (i) limx→1[x3−x2+1] (ii) limx→3[x(x+1)]
(iii) limx→−1[1+x+x2+…+x10].
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Step-by-Step Solution
Step 1: Evaluate the first limit
For a polynomial function, the limit as x approaches a certain value can be found by directly substituting that value into the function. In this case, we substitute x=1 into the expression x3−x2+1.
Step 2: Calculate the first limit
After substituting x=1, we perform the arithmetic operations: 13=1, 12=1. So the expression becomes 1−1+1, which simplifies to 1.
Step 3: Evaluate the second limit
Similarly, for the second limit, we have a polynomial function x(x+1). We can find the limit as x approaches 3 by directly substituting x=3 into the expression.
Step 4: Calculate the second limit
After substituting x=3, the expression becomes 3(3+1), which simplifies to 3(4), resulting in 12.
Step 5: Evaluate the third limit
The third limit involves a sum of powers of x, which is also a polynomial. We substitute x=−1 into each term of the sum.
Step 6: Calculate the third limit
When x=−1, the terms alternate between 1 and −1. Specifically, 1, (−1)1=−1, (−1)2=1, and so on. Since there are 11 terms in total (from x0 to x10), and pairs of (1−1) cancel out, the last term (−1)10 is 1, leaving a final sum of 1.